In this talk, we define some combinatorial objects which arise naturally in the study of link diagrams, the braid index of links and the other link invariants. These combinatorial objects are planar graphs equipped with some additional structure (edge colouring). We define reduction operations on these graphs in the context of calculating the braid index of links and the relationship between the braid index of links and the writhe of their diagrams. These operations generalize the similar ones introduced by Malesic and Traczyk. We also discuss the problem of optimal edge colouring, in our sense, the bipartite planar graphs and ask some open questions.
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